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Covering and Packing in Linear Space [chapter]

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2010 Lecture Notes in Computer Science  
Can one improve these bounds to only linear in the size of the family? Here, we answer the question in the affirmative regarding the space requirement, while not increasing the time requirement.  ...  When the size of the family is exponential in n, the fastest known algorithms for these problems use inclusion-exclusion and fast zeta transform, taking time and space 2 n , up to a factor polynomial in  ...  This bottleneck can be removed by taking a more balanced scheme with n 1 and n 2 about equal, yielding time and space O * (2 n/2 ) for each of the O * (2 n/2 ) processors.  ... 
doi:10.1007/978-3-642-14165-2_61 fatcat:udxf6oa5onclxk5p6yyrygxnxy

Covering and packing in linear space

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto
2011 Information Processing Letters  
Can one improve these bounds to only linear in the size of the family? Here, we answer the question in the affirmative regarding the space requirement, while not increasing the time requirement.  ...  When the size of the family is exponential in n, the fastest known algorithms for these problems use inclusion-exclusion and fast zeta transform, taking time and space 2 n , up to a factor polynomial in  ...  This bottleneck can be removed by taking a more balanced scheme with n 1 and n 2 about equal, yielding time and space O * (2 n/2 ) for each of the O * (2 n/2 ) processors.  ... 
doi:10.1016/j.ipl.2011.08.002 fatcat:rj32rkvxknefnerpdocpregzy4

Page 9329 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
This is a brief survey article in which the author reviews the main results, methods, and some open problems about simultaneous packing and covering in Euclidean n-space.  ...  One of the basic problems in the theory of packing and covering is to determine, for a given positive integer k, the minima!  ... 

Duality between packings and coverings of the Hamming space

G. Cohen, A. Vardy
2005 IEEE Information Theory Workshop, 2005.  
We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems.  ...  Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all  ...  Vice versa, if almost all codes in C (n, M + 1) are good packings, then almost all codes in C (n, M ) are bad coverings. Ã The same is true for linear codes.  ... 
doi:10.1109/itw.2005.1531849 dblp:conf/itw/CohenV05 fatcat:chqejnfnr5aqtpk43qiiqi7y3i

Duality between Packings and Coverings of the Hamming Space [article]

Gérard Cohen, Alexander Vardy
2005 arXiv   pre-print
We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems.  ...  Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all  ...  We are grateful to Alexander Barg and Ilya Dumer for helpful discussions. We are especially indebted to Ilya Dumer for sending us his proof in [6].  ... 
arXiv:cs/0507015v1 fatcat:6drgx35r5zhl3fq53ba6svds5u

Packing and covering in higher dimensions [article]

Gábor Fejes Tóth
2022 arXiv   pre-print
The present work surveys problems in n-dimensional space with n large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers.  ...  Subsequent results, such as the discovery of the Leech lattice and the linear programming bound, which culminated in the recent solution of the sphere packing problem in dimensions 8 and 24, were influenced  ...  The excellent book of Rogers [1964] gives an exhaustive account of packing and covering in high dimensions.  ... 
arXiv:2202.11358v1 fatcat:nekz37kvhnfmrbr3gsrfjushje

Packing And Covering Radii Of Linear Error-Block Codes

Rabiˆı DARITI, El Mamoun SOUIDI
2013 Zenodo  
The purpose of this paper is to generalize some results on the packing and the covering radii to the error-block case.  ...  We study their properties when a code undergoes some specific modifications and combinations with another code. We give a few bounds on the packing and the covering radii of these codes.  ...  In this paper, we extend the definitions of packing and covering radii to linear error-block codes.  ... 
doi:10.5281/zenodo.1088269 fatcat:jbus752rxnbw3pm6qayui65ixe

Packing and Covering δ-Hyperbolic Spaces by Balls [chapter]

Victor Chepoi, Bertrand Estellon
2007 Lecture Notes in Computer Science  
We consider the problem of covering and packing subsets of δ-hyperbolic metric spaces and graphs by balls.  ...  This result is established in the general framework of δ-hyperbolic geodesic metric spaces and is extended to some other set families derived from balls.  ...  In this note, we consider the problem of covering and packing by balls and union of balls of hyperbolic metric spaces and graphs.  ... 
doi:10.1007/978-3-540-74208-1_5 fatcat:w4kjwnfjjveppht6iu7o7tdvyu

Page 4168 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
88h:5 1021 51N Analytic and descriptive geometry See also 14053, 30021, 51017. 88h:51021 51N15 Somer, Josef (CS-CVUTE) Linear projective motions in spaces of finite dimension. (Czech.  ...  Kemnitz, Characterization of cen- trally symmetric convex domains in planes of constant curvature (pp. 179-189); A. Florian, Packing and covering with convex discs (pp. 191-207); Richard K.  ... 

Linear Programming Bounds for Codes via a Covering Argument

Michael Navon, Alex Samorodnitsky
2008 Discrete & Computational Geometry  
, on optimal packing of Hamming balls in a Hamming cube.  ...  We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument.  ...  We also thank Simon Litsyn and Madhu Sudan for valuable remarks.  ... 
doi:10.1007/s00454-008-9128-0 fatcat:ugze4mafhfgttm4uhf3p7jm5ua

Page 8987 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
a three-dimensional normed space, the problem of universal coverings for sets of unit diameter in a Euclidean space, and some related problems.”  ...  A solid packing of equal circles in either, the Euclidean plane or the sphere, is always the densest packing (similarly for coverings).  ... 

Page 4961 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
For a lattice L in an n-dimensional normed space X the packing and covering numbers are p(L) = 5 min{||x||: x € L \ {0}}, c(L) = max{dist(x, L): x € X}.  ...  (PL-LODZ) On the lattice packing-covering ratio of finite-dimensional normed spaces. Collog. Math. 59 (1990), no. 1, 31-33.  ... 

Linear programming bounds for codes via a covering argument [article]

Michael Navon, Alex Samorodnitsky
2007 arXiv   pre-print
We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument.  ...  In particular, we do not deal directly with Delsarte's linear program or orthogonal polynomial theory.  ...  We refer to [15, 11, 4, 14] for a detailed exposition of the notions discussed above, including error-correcting codes and their significance, packing in metric spaces, association schemes, Delsarte's  ... 
arXiv:math/0702425v1 fatcat:erel77czlnhlvpknoa5c3hatfi

Page 976 of Mathematical Reviews Vol. 35, Issue 5 [page]

1968 Mathematical Reviews  
Topics include convexity in topological linear spaces, packings and coverings, and convex bodies.  ...  Fejes Téth, Packings and coverings in the plane; L. Few, Multiple packing of spheres: A survey; W. J. Firey, Blaschke sums of con- vex bodies and mixed bodies; D.  ... 

Page 5022 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews  
In III- V many results on finite abelian groups are proved which imply tiling, packing and covering results for crosses and semicrosses.  ...  Gorshkova, Unsolved groups of motions in Davies spaces (pp. 55-62); A. I. Egorov, Maximally moving spaces of linear and hyperplanar elements (pp. 62-82); L. I.  ... 
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